Number Theory


Elementary Proof That There are Infinitely Many Primes p such that p − 1 is a Perfect Square (Landau's Fourth Problem)

Authors: Stephen Marshall

This paper presents a complete and exhaustive proof of Landau's Fourth Problem. The approach to this proof uses same logic that Euclid used to prove there are an infinite number of prime numbers. Then we use a proof found in Reference 1, that if p > 1 and d > 0 are integers, that p and p+ d are both primes if and only if for integer m: m = (p-1)!( 1/p + ((-1)^d (d!))/(p+d)) + 1/p + 1/(p+d) We use this proof for d = 2n + 1 to prove the infinitude of Landau’s Fourth Problem prime numbers. The author would like to give many thanks to the authors of 1001 Problems in Classical Number Theory, Jean-Marie De Koninck and Armel Mercier, 2004, Exercise Number 161 (see Reference 1). The proof provided in Exercise 6 is the key to making this paper on the Landau’s Fourth Problem possible.

Comments: 8 Pages.

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Submission history

[v1] 2017-03-09 09:40:06

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